We study the worst-case welfare of item pricing in the tollbooth problem . The problem is a special case of thecombinatorial auction in which (i) each of the $m$ items in the auction is anedge of some underlying graph . We show that the gap between the two settings is atleast a constant even when the underlying graph is a single path . In this setting where the tie-breaking power is on the buyers’ side, we also prove an $O(1)$ upper bound ofcompetitive ratio for special families of graphs . Inparticular, we improve the ratio when the input graph $G$ is a tree, from 8(proved by Cheung and Swamy) to 3 . We prove that the ratio is $2$ (tight) when$G$ was a cycle and $O(\log^2 m)$ when $G) is an outerplanar graph . All positive results above require that the seller can choose a propertie-breaking rule to maximize the welfare of the welfare to maximize their welfare in this setting . We also prove the ratio was $2 ($1) when $2) when the output graph was a cyclical and $1.1/4/1/2/4 . We conclude that the outcome of the outcome is $1/3/5/4 is the same for $2/5.5/5 . We say that the buyer can choose the seller

Author(s) : Zihan Tan, Yifeng Teng, Mingfei Zhao

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Keywords : graph - welfare - ratio - prove - case -

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